What is the relationship between points and neighborhoods in topology?

[Orion1]

What IS Infinity?[/color]

Many of us might consider numbers the most sure-footed way to come within sight of infinity, even if the mathematical notion of infinity is something we'll never even remotely comprehend.

Mathematicians tell us that any infinite set?anything with an infinite number of things in it?is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is?you got it?infinity. And yet, there is always an even larger number: infinity plus 1 is not larger than infinity, but 2^infinity is.

When we're told that the decimals in certain significant numbers, like pi and the square root of two, go on forever, we can somehow accept that, especially when we learn that computers have calculated the value of pi, for one, to over a trillion places, with no final value for pi in sight. (For more on pi, see Approximating Pi.) When we're told that there are 43,252,003,274,489,856,000 possible ways to arrange the squares on the Rubik Cube's six sides, we may feel intuitively (if not rationally) that we must be on our way to the base of that loftiest of all peaks, Mt. Infinity.

One reason we may feel this way is that such numbers are as intellectually unapproachable to the mathematically challenged as infinity itself. Take a Googol. A Googol is 10100, or 1 followed by 100 zeroes, and is the largest named number in the West. The Buddhists have an even more robust number, 10140, which they know as asankhyeya. Just for fun, I'll name a larger number yet, 101000. I'll call it the "Olivian," after my daughter. Now, doesn't an Olivian get me a little closer to infinity than the Googolians or even the Buddhists can get? Nope. Infinity is just as far from an Olivian as it is from a Googol?or, for that matter, from 1.

Infinities do come in two sizes, of course?not only the infinitely large but also the infinitely small. As Jonathan Swift wrote, "So, naturalists observe, a flea/Has smaller fleas that on him prey/And these have smaller still to bite `em/And so proceed ad infinitum."

Of course, just when we think we have infinity in the palm of our hands, we watch it evaporate in the harsh light of another of those confounding paradoxes: the numerals 2 and 3 are separated by both a finite number (1) and an infinity of numbers. This conundrum spawned one of the great paradoxes of history, known as Zeno's paradox. Zeno was a Greek philosopher of the fourth century B.C. who "proved" that motion was impossible. For a runner to move from one point to another, Zeno asserted, he must first cover half the distance, then half the remaining distance, then half the remaining distance again, and so on and so on. Since this would require an infinite number of strides, he could never reach his destination, even if it lay just a few strides away.

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.

We believe Hamlet when he says "I could be bounded in a nutshell/And count myself a king of infinite space." The shortest length physicists speak of is the Planck length, 10^-33 centimeters. But might not there be an even shorter length, say, 10^-333 centimeters, or 10-an infinite number of 3's centimeters?

as one mathematician pointed out to me, infinity is an abstract concept, appearing only in our mental images of the universe. It is not actually in the universe.

The Greeks, in fact, invented apeirophobia, fear of the infinite. (The term comes from the Greek word for infinity, apeiron, which means "without boundary.") Aristotle would only admit that the natural numbers (1, 2, 28, etc.) could be potentially infinite, because they have no greatest member. But they could not be actually infinite, because no one, he believed, could imagine the entire set of natural numbers as a finished thing. The Romans felt just as uncomfortable, with the emperor Marcus Aurelius dismissing infinity as "a fathomless gulf, into which all things vanish."

The ancients' horror infiniti held sway through the Renaissance and right up to modern times. In 1600, the Inquisitors in Italy deemed the concept so heretical that when the philosopher Giordano Bruno insisted on promulgating his thoughts on infinity, they burned him at the stake for it. Later that century, the French mathematician Blaise Pascal deemed the concept truly disturbing: "When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there ... now instead of then." Martin Buber, an Israeli philosopher who died in 1965, felt so undone by the concept of infinity that he "seriously thought of avoiding it by suicide."

...like that engendered by this gem from another anonymous sufferer of our common infirmity: "Infinity is a floorless room without walls or ceiling."

Reference:
http://www.pbs.org/wgbh/nova/archimedes/contemplating.html
http://www.pbs.org/wgbh/nova/archimedes/infinity.html

 

[wuliheron]

Words and concepts only have demonstrable meaning according to their function in a given context. Outside of a given context, infinity is just as meaningless as any other concept.

Within any context, it is a notably vague term. When I talk about a chair, I can point to an example and ramble on for days about exactly why it is called a chair, pointing out each and every feature and their uses, which you are free to examine for yourself. Although people certainly ramble on for days about infinity, no one has ever been able to prove infinity exists as more than an idea.

That is not to say it is a useless fantasy. Infinity possesses the properties of both the demonstrable and undemonstrable. It is a cross between the utterly paradoxical and rational. Even the paradoxical has its uses, and being a bit less extreme infinity has even more obvious uses. However, in and of itself infinity is demonstrably useless, it is only a useful concept within the context of the finite. Here is an ancient chinese poem which expresses such relationships.

Tools

Thirty spokes meet at a nave;
Because of the hole we may use the wheel.
Clay is moulded into a vessel;
Because of the hollow we may use the cup.
Walls are built around a hearth;
Because of the doors we may use the house.
Thus tools come from what exists,
But use from what does not.

 

[Hurkyl]

A Googol is 10100, or 1 followed by 100 zeroes, and is the largest named number in the West.

Wee, math trivia! We actually have a number called a googolplex which is a 1 followed by a googol zeroes.

 

[arildno]

Besides, the Ramsey number is a LOT bigger than either..

 

[Smurf]

Wee, math trivia! We actually have a number called a googolplex which is a 1 followed by a googol zeroes.

I had trouble comprehending a googol, now you go and throw this at me

-Ruler of the Universe,
Smurf

 

[metacristi]

Wee, math trivia! We actually have a number called a googolplex which is a 1 followed by a googol zeroes.

Still not enough to win the 'who can name the bigger number' contest :-)

http://www.cs.berkeley.edu/~aaronson/bignumbers.html

 

[NeutronStar]

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.

This is actually a huge misconcpetion on the part of the mathematical community. Calculus does not resolve Zeno's paradoxes. For two reasons,...

First, if we exam the defintion of the limit in detail it never actually says that any of those infinite processes actually equal the value of the limit. It only says that as some variable approaches some value then a particular function or process approaches infinity.

Well, gee, that's what Zeno was saying all along! So how is that a solution?

In fact the definition of the limit is clearly set up as to not permit the variable from actually reaching the value in question. In other words, there exists a delta greater than zero 0 such that,... In other words, the calculus limit says absolutely nothing about what happens should we decide to consider delta=0. The definition of the limit is no longer valid in that form. This prevents us from actually claiming that the value is ever actually reached. This is why good mathematicians are sure to say things like "in the limit" a value is equal to something. Because to actually say flat out that it is equal is simply wrong. It denies the very conditions of the definition of the limit. Unfortunately far too many mathematicians seem to have dropped the "in the limit" phrase and think that something is actually equal to a calculus limit if they can prove that a limit exists. It is not. That is actually a misuse of the definition of a limit.

Secondly, Zeno's real question is this,... "How can it be that an infinite number of tasks can be completed". Nowhere in calculus is it ever claimed that an infinite number of processes can be completed. In fact, to prove a limit all we need to do is prove things like boundedness and trends. If we look at all of the definitions and proofs for any limit we will clearly see that we haven't proven anywhere along the way that we have actually completed an infinite process, nor have we proven that it can be completed. All we have done is shown that no matter how long we continue the process we will continue to get closer and closer to the value that we call the limit.

Well, again, Zeno would say, "So what? That's what I've been saying all along!"

The mathematical community is absolutely incorrect to claim that it has solved Zeno's paradoxes with calculus yet this is a widely held misconception. The calculus limit really doesn't say any more than Zeno had alread pointed out. If he were alive today I am absolutely certain that he would simply say that these people just haven't truly understood the question that he is asking,... He wants to know how an infinite number of tasks ever be completed? Calculus does NOT answer that question, nor does it claim to. Yet it still claims to have solved Zeno's paradox. Clearly the mathematical community doesn't understand the question that Zeno is asking.

Fortunately physics has answered Zeno's paradox. Time and space are not continuous, they are quantized. Zeno was probably right. If time and space were continuous we probably wouldn't be able to move. But since they are quantized only need to complete a finite number of tasks to move and finite distance because we do it in quantum jumps skiping over the supposed continuum.

After all, Zeno's paradox is only a paradox in a continuous universe. Once the universe is known to be quantized it's no longer a paradox as to why we can move. Calculus has nothing at all to do with the paradox at all.

 

[robert Ihnot]

Orion1: It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit.

As Neutron Star points out this is commonly supposed. It is a way for calculus buffs to indulge in self-congratulations. But, probably every generation could find some answer to Zeno.

However, I do believe that Archimedes was aware of the finite limit of the infinite sum. Indeed he used "infinite triangle sums" to discover the area of parabola.

 

[Hurkyl]

Fortunately physics has answered Zeno's paradox. Time and space are not continuous, they are quantized.

Physics has said no such thing.

No empirically verified theory exhibits any sort of geometric quantization. While some of the promising research topics do exhibit geometric quantiziation, they still don't exhibit quantization of position.

 

[NeutronStar]

Physics has said no such thing.

No empirically verified theory exhibits any sort of geometric quantization. While some of the promising research topics do exhibit geometric quantiziation, they still don't exhibit quantization of position.

I beg to differ.

It is meaningless to speak in terms of absolute position. That might be an abstract mathematical concept, but it has no meaning with respect to the universe.

In our universe any "real" particle or phenomenon exhibits some form of energetic disturbance. If it didn't how could we even claim that it exists? How could we even know of such a "invisible" entity. It would be completely undetectable in any possible way.

Most particles in the universe exhibit as least some mass which associates them with gravitational energy. Any known massless particle (like say a photon) would is associated with some other form of energy (like maybe electromagnetism)

So in any case, every particle that can possibly be associated with a position must necessarily take on a quantized position relative to other particles. In other words, it can only move in a way to take on, or give up, a quantum of energy.

Therefore, any "real" concept of position in the universe is necessarily a quantized concept due to an objects potential energy relative to other objects.

It's true that we can fool ourselves into believing that there are somehow "abstract" possible positions between these particles that aren't quantized. But isn't that putting something onto the universe that isn't really there? I mean, what good is a notion of a position that no particle can take on?

So I do hold that physics has shown that position is necessarily quantized in the real universe, and it is totally meaningless to talk about some kind of continuum when the particles in the universe do not actually behave in that fashion. Why make things up that aren't a true reflection of the properties of our universe?

It's going to take people a very long time to let go of this idea of the continuum. Our universe just isn't continuous. That's all there is to it. At the quantum level things "jump" around, They just don't move continuously. They also don't traverse the space between their jumping. There are either here or there, but never in between. That's the true nature of our quantized universe.

This isn't just true of things that are bound to energy levels within atoms. Anything that changes position in the universe is changing its relative position to something else. Therefore it is changing the relative potential energy whether it be gravitational energy or some other form. Therefore position is quantized for everything in the universe.

One could argue that a so-called "free electron", or whatever, is not restrained to quantized motions. But for the reasons I just gave an electron is never actually "free". Also, would it really even be meaningful to talk about the position of such a free particle? To speak of its position we can only do so in relative terms. Once we have set its position relative to something then in a very real sense it is bound to that reference point by mere convention and can only change its potential energy relative to that position in a quantized fashion.

So how can the idea of position ever not be quantized in our universe?

 

[Hurkyl]

It is meaningless to speak in terms of absolute position.

Even so, you can still speak about position being quantized.


But I should have been more precise: what I meant to say is that length is not quantized.


Energy is also not quantized. It is true that bound particles exhibit quantized energy levels, but energy, in general, can come in any quantity.


P.S. don't forget that there is no quantum theory of gravity. Furthermore, TMK, there has only been one experiment, ever, that has demonstrated quantum effects in a gravitationally bound system. (Of course there has only been one experiment that has tested it)

 

[anti_crank]

So in any case, every particle that can possibly be associated with a position must necessarily take on a quantized position relative to other particles. In other words, it can only move in a way to take on, or give up, a quantum of energy.

How does this follow? What is this quantum of energy, and where does it come from?

 

[NeutronStar]

P.S. don't forget that there is no quantum theory of gravity. Furthermore, TMK, there has only been one experiment, ever, that has demonstrated quantum effects in a gravitationally bound system. (Of course there has only been one experiment that has tested it)

Oops! You're right. I forgot they didn't get there yet.

But I think it's pretty obvious that this will have to be the case. I mean, do we think that quantum theory will end up giving up its quantized nature in order to agree with GR. Or is it more likely that GR will have to give up its smooth continuum to agree with QM?

 

[Hurkyl]

I won't pretend to know what quantum geometry looks like. I will just comment that the amount of discretization inherent in quantum mechanics is vastly exaggerated in popular accounts.

 

[Canute]

NeutronStar

I thought what you said about the notion of infinitessimals and limits not solving Zeno's paradoxes was spot on. Especially:

"The mathematical community is absolutely incorrect to claim that it has solved Zeno's paradoxes with calculus yet this is a widely held misconception. The calculus limit really doesn't say any more than Zeno had alread pointed out."

However you go on to say:

"Fortunately physics has answered Zeno's paradox. Time and space are not continuous, they are quantized. Zeno was probably right. If time and space were continuous we probably wouldn't be able to move. But since they are quantized only need to complete a finite number of tasks to move and finite distance because we do it in quantum jumps skiping over the supposed continuum."

I see this as the reverse of the truth for two reasons. Firstly, I can't agree that science has shown that time and space are quantised. How has science done this?

Second, Zeno's point was that if one takes spacetime to be quantised, or, equivalently, takes the number line to be a series of points, then motion is (mathematically) paradoxical. If one takes spacetime as a continuum then the paradox disappears. This is consistent with the effectiveness of the calculus, since a 'point' in a continuum is actually an infinitely divisible range. By that I mean, the only thing that is infinitewly divisible is a point in a continuum. If spacetime (or the number line) is quantised then why do we need to use infinitessimals to calculate motion?

If you bring Zeno up to date and imagine Achilles and the Tortoise to be two particles, each being one fundamental quanta in diameter, moving in a spacetime that is quantised into fundamental quanta of time and space (discrete moments and discrete positions), then relative motion between A and T makes no sense at all. That is, not unless you allow length contraction and time dilation.

 

[selfAdjoint]

If you bring Zeno up to date and imagine Achilles and the Tortoise to be two particles, each being one fundamental quanta in diameter, moving in a spacetime that is quantised into fundamental quanta of time and space (discrete moments and discrete positions), then relative motion between A and T makes no sense at all. That is, not unless you allow length contraction and time dilation.

Well you could define position change; you observe particle T at time 0 and again later at time t, and get different postions, and likewise for particle A.

 

[NeutronStar]

Zeno's point was that if one takes spacetime to be quantized, or, equivalently, takes the number line to be a series of points, then motion is (mathematically) paradoxical.

Yes, that is true if it is also assumed that a finite line segment contains an infinite number of points. Which is what was believed in Zeno's day. And is still believed to be true today. But actually that can't possibly be the case. That situation is a logical contradiction. That is to say that such a concept is a logical inconsistency.

If one takes spacetime as a continuum then the paradox disappears.

I'm afraid I don't see why it would disappear if spacetime is a continuum. How does this change Zeno's paradox or resolve it?

This is consistent with the effectiveness of the calculus, since a 'point' in a continuum is actually an infinitely divisible range. By that I mean, the only thing that is infinitely divisible is a point in a continuum. If spacetime (or the number line) is quantized then why do we need to use infinitesimals to calculate motion?

I'm afraid that you've lost me here.

How can a 'point' be an infinitely divisible range? I can see the space between points as being an infinitely divisible range. But what sense does it make to say that any given point itself is an infinitely divisible range?

And more to the point (no pun intended) where in the formalism of calculus is there any mention of a point being an infinitely divisible range, or that such a concept has anything at all to do with infinitesimals?

It's not in the formal definition of the limit I can say that much with certainty.

 

[StatusX]

Why can't we complete an infinite number of tasks? This is an assumption. If it is taken true, and a "task" is defined as anything that you can describe in words, such as "going half the distance left," then Zeno has a point. If not, then his argument is meaningless. What reasons do you have for believing it?

 

[Canute]

Well you could define position change; you observe particle T at time 0 and again later at time t, and get different postions, and likewise for particle A.

True. But try working out the relative motion one instant at a time. Particle A can go no faster than one quanta of distance per instant (otherwise it would be in more than one place at the same time) and particle B can go no slower than one quanta of distance per instant, (otherwise it would be stationary). The only solution is to allow A to be at more than one place at a time (changing its length) or for the instants of A to be longer than the instants of B (changing its clock). I can't see a way around this problem except to say that spacetime is not quantised. This doesn't seem to disagree with any evidence as far as I can tell. (We seem to having this discussion in two threads so sorry if I'm repeating stuff).

 

[Canute]

Yes, that is true if it is also assumed that a finite line segment contains an infinite number of points. Which is what was believed in Zeno's day. And is still believed to be true today. But actually that can't possibly be the case. That situation is a logical contradiction. That is to say that such a concept is a logical inconsistency.

I agree. However the problem does not arise if the number line is conceived of as a continuum (as Charles Sanders Peirce argued it should be).

I'm afraid I don't see why it would disappear if spacetime is a continuum. How does this change Zeno's paradox or resolve it?

It allows points to be treated as if they are infinitely divisible ranges, and thus they can be treated non-paradoxically by mathematics. An infinitely divisible point is by definition not a point but a range. If it was a mathematical point it would not be divisible.

How can a 'point' be an infinitely divisible range? I can see the space between points as being an infinitely divisible range. But what sense does it make to say that any given point itself is an infinitely divisible range?

You're right, a point cannot be a range. However an infinitely divisible point (or rather, the concept of an infinitely divisible point) is not a point.

And more to the point (no pun intended) where in the formalism of calculus is there any mention of a point being an infinitely divisible range, or that such a concept has anything at all to do with infinitesimals?

There's no mention of it because it's taken for granted that such points have extension (conceptually at least). If they did not then how could they be divided?

It's not in the formal definition of the limit I can say that much with certainty.

A limit is defined according to some specified tolerance of error. That is, at the limit .9999... equals 1. But only because we decide to stop calculating and accept some innacuracy.

 

[matt grime]

A limit is defined according to some specified tolerance of error. That is, at the limit .9999... equals 1. But only because we decide to stop calculating accept some innacuracy.

Oh, dear, I cannot begin to explain the inaccuracies there, they all tumble to get out of my head at the same time.

 

[Canute]

Oh, dear, I cannot begin to explain the inaccuracies there, they all tumble to get out of my head at the same time.

Yes, I knew I was pushing my luck it a bit there. But it's not very helpful to me if you just throw up your hands. What was wrong with what I said?

 

[matt grime]

What are the real numbers? They are by definition the complete totally ordered field, or the set of dedekind cuts of the rationals or the set of equivalence classes of convergent cauchy sequences of rationals or a model of one of these, or possibly many other equivalent things. In any case 0.99... =1 is a simple consequence of the axioms. It has nothing to do with approximating, or adding things up and approaching and so on. These are formal mathematical objects, this is not a constructive universe, what you wrote just isn't mathematical.

If I gave in it's simply because this is such a common problem that I can't believe you haven't come across many of the threads on it. All objections are based upon the fact that the objector doesn't know the mathematics of the real numbers presuming them to be something they are not.

 

[NeutronStar]

The limit is defined according to some specified tolerance of error.

Gee, the formal definition of the calculus limit that I was taught has nothing in it at all about any specified tolerance of error. That would be the formal Weierstrass delta-epsilon definition. There's nothing in that definition about any tolerance of error.

That is, at the limit .9999... equals 1. But only because we decide to stop calculating and accept some innacuracy.

This is perfectly true. "In the limit" 0.999999... = 1. But that is not the same thing as saying that 0.9999... = 1.

To say that the 0.9999... = 1 outside of the context of the limit is absurd. And anyone who fully understands the Weierstrass delta-epsilon definition of the limit knows that delta can never be said to be zero. Therefore to say that 0.9999...=1 in any asbsolute sense is to simply deny the very defintion of the limit. Weierstrass was very careful to include in his definition:

\exists\delta\ni\delta>0

Yet so many mathematicians seem to claim that deta can actually be thought of as having been taken completely to zero. They're simply wrong. They are ignoring Weierstrass's definition. They are jumping to conclusions that simply aren't in the formal definition.

By the way, Weierstrass didn't put that condition into his definition of the limit just for fun. He realized that the whole definition would break down if that condition isn't included! It's an extremely important concept to the whole idea of the limit. Yet mathematicians consistently try to ignore it! It just goes to show that they don't fully grasp the concept I suppose.

Any mathematician who tries to claim that 0.9999,...=1 outside of the concept of the limit should be totally embarrassed. There's just no justification for it. It's not a mathematical idea.

More to the point, 0.9999,... does not equal 1. However, the limit of 0.9999... does equal 1. See what I mean? L=1 The limit equals 1.

0.9999... doesn't equal 1. Its limit equals 1. That's a whole different concept than saying that 0.9999... = 1. In fact, to say that latter is simply mathematically incorrect. There's no basis for it. It's simply incorrect.

So I actually agree with you Canute. We do decide to stop calculating. In fact all we really need to do to satisfy Weierstrass's limit definition is to prove trends actually. All that the limit really says is that 0.999999... is approaching 1. It never claims in any way that it ever actually equals 1. That is an incorrect conclusion. I simpy can't undestand why so many mathematicians insist on jumping to this conclusion. It really makes me wonder who they had as a teacher?

 

[matt grime]

They had mathematicians as teachers, Fields Medal winning mathematicians. So, Neutron, what is the definition of Real Numbers you are using and what is the definition of 0.99... you are using?

0.999... is approcahing 1? 0.999... is a fixed real number, it is not itself approaching anything in this active sense.

Two real numbers x and y are equal iff, for all d>0 |x-y| < d.

Proof: Suppose not, if two real numbers are not equal that x-y=r some r. Then d=r/2 will suffice. The reverse implication is trivially left as an exercise for the reader.

By the definition of the reals, I need only consider d in the rationals.

So 1- 0.999... < 1/10^n for all n clearly (it is larger than all its partial sums) hence 1=0.999...

(the reals are the space where all convegent cauchy sequences of rationals have a limit).

 

[NeutronStar]

Why can't we complete an infinite number of tasks? This is an assumption.

Unless you believe that an infinite number is finite how could it ever be completed?

To claim that an infinite number of tasks can be completed is to simply deny the meaning of "infinite".

All you would be saying is that infinity is finite. If you can live with that more power to you. It doesn't make any sense at all to me .

Intuitively that would be like saying that an endless process can come to an end. Well, if it can come to an end then why did we call it an "endless" process to begin with? We must have been wrong in the first place!

It's simply a logical contradiction in terms to claim that an infinite process can be completed. It makes no sense.

 

[NeutronStar]

(the reals are the space where all convegent cauchy sequences of rationals have a limit).

Well, there you go. You just proved that I'm correct.

The very space that you are calling the "reals" is defined on the concept of the Weierstrass delta-epsilon definition of the calculus limit via the Cauchy sequences of rationals.

Therefore anything that any mathematician ever says about the real numbers should be preceded by the phrase, "In the limit".

That's all I ask because I fully understand the concept of the calculus limit and as long as you precede all of your statements about real numbers with the phrase "In the limit" I have no problem because I know what that means.

So as far as I can see you've just inadvertently agreed with me.

 

[StatusX]

Unless you believe that an infinite number is finite how could it ever be completed?

To claim that an infinite number of tasks can be completed is to simply deny the meaning of "infinite".

All you would be saying is that infinity is finite. If you can live with that more power to you. It doesn't make any sense at all to me.

What kind of argument is this? You are still assuming an infinite number of "tasks" can never be completed. Your saying you can't complete an infinite number of tasks because to complete a number of tasks, that number must be finite, and so that would mean infinity is finite, which is a contradiction. That is absurd reasoning. If the tasks take less and less time, such that the infinite sum of the times of the tasks converge, then what possible reason is there to believe you couldn't do them?

As for what you said about 0.999... only equalling 1 in the limit, I have to ask you, the limit of what? I assume you mean the limit as n goes to infinity of:

\sum_{k=1}^{n} 9 \cdot 10^{-k}

But what you are mistakingly assuming is that 0.999... represents, in some sense, the process of taking this sum. This is unclear, and I don't know how you could prove anything about such a poorly defined concept. What you have to realize is that the definition of 0.999... is:

0.999... \equiv \lim_{n \rightarrow \infty} \sum_{k=1}^{n} 9 \cdot 10^{-k}

I'm pretty sure you already conceded this is 1. If you disagree with this definition, then you are not using the same notation as the rest of us, plain and simple.

Be careful before you assume you know something that professional mathematicians don't. You call their careful work of centuries "simply incorrect" because you don't understand it, and its ironic that you think theyre the ones who should be "totally embarassed."

 

[NeutronStar]

You call their careful work of centuries "simply incorrect"

Whoa!

I never said any such thing!

I believe that most serious professional mathematicians will agree that any conclusions that are based on the definition of the limit should be preceded by the phase "in the limit".

Any mathematician worth his salt will quickly point out that saying that the limit of f(x) at c is equal to L does NOT automatically imply that f(c)=L.

That *is* mathematics! To claim otherwise is "simply incorrect" not according to me, but according to mathematical formalism period amen.

Now is there any mathematician who really wants to argue with that?

 

[StatusX]

0.9999... doesn't equal 1. Its limit equals 1. That's a whole different concept than saying that 0.9999... = 1. In fact, to say that latter is simply mathematically incorrect. There's no basis for it. It's simply incorrect.
...
All that the limit really says is that 0.999999... is approaching 1. It never claims in any way that it ever actually equals 1. That is an incorrect conclusion. I simpy can't undestand why so many mathematicians insist on jumping to this conclusion. It really makes me wonder who they had as a teacher?

What you called incorrect is actually correct. You said some pretty derogatory things about people who understand something that you don't.

Any mathematician worth his salt will quickly point out that saying that the limit of f(x) at c is equal to L does NOT automatically imply that f(c)=L.

No doubt about this, but that's not what you said.

 

[synergy]

summary?

I just realized I only read page 1.
So,
continuum => infinitely divisible => an infinite number of tasks and
discrete => A and B always are the same speed or stationary. This is, at least, what I think Canute was saying:
Quote: Canute said:
try working out the relative motion one instant at a time. Particle A can go no faster than one quanta of distance per instant (otherwise it would be in more than one place at the same time) and particle B can go no slower than one quanta of distance per instant, (otherwise it would be stationary). The only solution is to allow A to be at more than one place at a time (changing its length) or for the instants of A to be longer than the instants of B (changing its clock). I can't see a way around this problem except to say that spacetime is not quantised. This doesn't seem to disagree with any evidence as far as I can tell.
end of quote.
It still sounds like a paradox to me, perhaps a juxtaposition of states could lead to a loophole? Space is both discrete and continuous at the same "time"?
Aaron

 

[StatusX]

The individual particles can only "travel" at one speed, but patterns of them are more flexible. For example, the game of life is a sort of quantized universe, and this page mentions how different "speeds" are possible in this setting. They suggestively call the maximum speed of 1 cell per unit time "c."

http://www.ericweisstein.com/encyclopedias/life/Spaceship.html

 

[jcsd]

Whoa!

I never said any such thing!

I believe that most serious professional mathematicians will agree that any conclusions that are based on the definition of the limit should be preceded by the phase "in the limit".

Any mathematician worth his salt will quickly point out that saying that the limit of f(x) at c is equal to L does NOT automatically imply that f(c)=L.

That *is* mathematics! To claim otherwise is "simply incorrect" not according to me, but according to mathematical formalism period amen.

Now is there any mathematician who really wants to argue with that?

We're talking about the limit of sequences in the rationals,, that limit is not guranteed to be rational, however it is guraenteed to correspond to a real number, by the defitnion of a real number! Your point is irrelvean as we're tlaking boaut the defintion of the reals so we have no need of the preface 'in the limit' as we know that the limit will exists as we use that limit to define the reals.

 

[NeutronStar]

What you called incorrect is actually correct. You said some pretty derogatory things about people who understand something that you don't.

Well, it would only be taken as derogatory by those to which it is applicable.

So let me understand you. You are saying that modern mathematical formalism says that an infinite number of tasks can be completed without referencing Weierstrass's definition of the limit which clearly does not support this conclusion.

I'm open to that. I've just never heard of that part of the formalism. Clearly any reference to the real numbers automatically references Weierstrass's limit definition since the reals are defined upon that concept so it can't have anything to do with real numbers.

 

[jcsd]

We're not saying that we're carrying an infinite amount of tasks we're just defining the reals in terms of limits, infact you can certainly define all the algebraic numbers without ever mentioning a limit, for example the set {x in Q|x>0 and x^2 > 3} defines sqrt(3) without a mention of a limit.

 

[StatusX]

actually limits never invoke the concept of infinty. A limit is defined in purely finite terms. to jump from saying that the limit of a continuous function f(x) as x goes to c is L to saying that f(c)=L would be invoking the idea of infinity. In fact, a function is DEFINED to be continuous at c if this happens to be true. This captures the intuitve idea of infinity without explicitly using it. Like I said before, 0.999... is DEFINED as the limit of that sequence of sums, and infinity is never involved.

 

[NeutronStar]

Infact you can certainly define all the algebraic numbers without ever mentioning a limit, for example the set {x in Q|x>0 and x^2 > 3} defines sqrt(3) without a mention of a limit.

Well, yes. But this rests on the assumption that finite intervals are continuums. This is an assumption that I must confess that I am probably alone in not accepting. However, I have never seen a convincing proof that finite intervals are continuums. On the contrary I can show several solid logical arguments why they can't be.

actually limits never invoke the concept of infinty. A limit is defined in purely finite terms.

I absolutely agree with this.

to jump from saying that the limit of a continuous function f(x) as x goes to c is L to saying that f(c)=L would be invoking the idea of infinity. In fact, a function is DEFINED to be continuous at c if this happens to be true.

Yes, but don't confuse the word continuous with the word continuum. This state of affairs does not imply a continuum in any way. As a matter of fact, by Weierstrass's strict definition of the limit this particular definition of "continuous" necessarily implies a quantized situation. This is true because we are not permitted to consider delta to ever be equal to zero. Therefore we must necessarily view this definition of "continuous" as being quantized.

So to mathematically say that a function is continuous (by this definition) does not in any way imply that it defines a continuum. In fact, it actually implies that it must necessarily be discrete (or quantized).

 

[jcsd]

Well, yes. But this rests on the assumption that finite intervals are continuums. This is an assumption that I must confess that I am probably alone in not accepting. However, I have never seen a convincing proof that finite intervals are continuums. On the contrary I can show several solid logical arguments why they can't be.

The proof that it defines a continuum is the fact that it defines the real numbers, this is trivially true as the continuum is the set of real numbers!

Yes, but don't confuse the word continuous with the word continuum.

I think you've yuor terminology mixed up here, in the most general sense a continuum is a set with some sort of order realtion something like the real line i.e. has the cardianlity of the continuum and between any two numbers there is another number.

We're dealing with sequences which are functions of the type f:N->R so the epilson-delata defitnion is irrelvant.

This state of affairs does not imply a continuum in any way. As a matter of fact, by Weierstrass's strict definition of the limit this particular definition of "continuous" necessarily implies a quantized situation. This is true because we are not permitted to consider delta to ever be equal to zero. Therefore we must necessarily view this definition of "continuous" as being quantized.

No it does not imply it is quantized, the fact that we don't want delat ito eb equal to zero is that we are taking the limit of the function at a point, so we must ignore what is going on at that point.

So to mathematically say that a function is continuous (by this definition) does not in any way imply that it defines a continuum. In fact, it actually implies that it must necessarily be discrete (or quantized).

It does imply that the function is not discrete and as I said anyway this defitnion os irrelvant to whta is being talked about.

 

[Hurkyl]

There is no logical problem with a continuum made up of individual points, but it is true that ranges play an important part of the concept. A topology is made up of two things: points, and neighborhoods. For the real line, we can indeed take the neighborhoods to be "ranges". More specifically, the neighborhoods can be taken to be the open intervals.


.999... is a number, not a sequence of things for which you need to take a limit to get a number.

It's simply a logical contradiction in terms to claim that an infinite process can be completed. It makes no sense.

You're certainly free to have the opinion that it makes no sense, but when you call something a logical contradiction you should demonstrate it.

 

[NeutronStar]

You're certainly free to have the opinion that it makes no sense, but when you call something a logical contradiction you should demonstrate it.

Well, I must confess that I could never prove it to anyone who believes that a point is range. My entire proof requires that a point be dimensionless.

I must be getting old because I was always taught that a mathematical point is dimensionless. I wasn't aware that they've become engorged over the years. What was the purpose of that? Who engorged them?

I thought they were pretty cool concepts when they were dimensionless.

In any case, I would like to point out that these recent developments in mathematics can hardly be called "The careful work of centuries". I'm sure that Euclid would roll over in his grave if he knew that his dimensionless points had gone off their diets.

 

[StatusX]

The fact is, continuity in the sense I was referring to is only a valid concept for functions from R to R. Discrete functions cannot possibly be continuous, because it would be impossible for the limit to exist. That's the point. You can pick ANY e>0, and if the function is continuous, there will be a d>0 such that |f(c-d)-f(c)|<e. This is not possible if the function is quantized, because if e<|f(c-q)-f(c)|, where q is the size of the quanitization, then there will be no d that will work. So I wasn't confusing continuum with continuous, but they are more closely related than you seem to think.

Actually, I'm not even sure what were talking about. Whether 0.999... is 1? Whether the real numbers exist? If infinite tasks are impossible? Which is it?

 

[Canute]

Seeing as how it was my offhand laymans remark about .99... equaling 1 that set this off I'll clarify what I meant about points and ranges. I'm well aware that I'm not a mathematician, but for me this is more about meta-mathematics than mathematics per se. I expect some of this is incorrect but I'll say my piece and you can tear it apart.

A mathematical point is by definition dimensionless. How then can such a point be divisible? If one models motion as taking place in a medium made up of points which are each infinitely divisible then this in fact models motion in a continuum, not quantised motion. Thus, for practical purposes, the calculus gets around Zeno's objections to motion in quantised spacetime by un-quantising it. However it is a fudge, since the calculation of these points uses the concept of limits. It defines points in spacetime as infinitely divisible but then treats them as if they are not.

When I said .999...=1 I should have said 'in the limit'. I was suggesting that in reality .999... does not equal 1, it equals .999... Obviously this number, as it expands, approaches 1. However it never becomes 1, precisely because if points are infinitely divisible then there is always a number between .999... and 1. To round off .999... to 1 is to assume that points are not infinitely divisible.

I was suggesting that the concept of infinitessimals does not solve Zeno's paradoxes because in the calculus one has ones cake and eats it. Points are considered to infinitely divisible but they are not infinitely divided. In the end isn't the whole purpose of the calculus the avoidance of infinite divisions?

Thus the concept of infinitessimals allows us to model motion mathematically, but does not answer the question of how motion is possible if spacetime is quantised.

To put it another way, infinitessimals are conceptual things, mathematical tools, not things that exist. If spacetime is quantised then its fundamental quanta have physical extension. As such they are not points but ranges, the extent of the range determined by the diameter of the point. When we divide it again and again the range is reduced, but it cannot be reduced to nothing except conceptually. (The Dedekind Cut seems relevant here but I won't risk saying anything about about that).

That's not very clear but best I can do at the moment.

 

[matt grime]

It is incorrect to say that "in the limit 0.999...=1" as a piece of English, since 0.999... is itself a limit point so you're being tautogical.

It is correct to say that the limit as n tends to infinity of the n'th partial sum is 1, it also correct to say 0.999...=1 since we have implicitly constructed it as a limit, and they are in the same equivalence class of cauchy convergent sequences and hence in the space of reals they are equal. (Better to say equivalent perhaps, but equal is the norm).

You are saying that we must always say "in the limit the limit of the partial usm of 0.99... is 1" which is completely unnecessary.

 

[NeutronStar]

Constructing Mathematical Objects from Limits?

It is correct to say that the limit as n tends to infinity of the n'th partial sum is 1, it also correct to say 0.999...=1 since we have implicitly constructed it as a limit.

Constructed it as a limit?

This is the kind of talk that just blows me away.

Where's the justification for claiming that anything has been "constructed"?

This isn't a personal attack on you Matt. I realize that many mathematicians have been taught this. But who's teaching this and why? Who decided that a limit can be said to have "constructed" something? Where did that idea come from? Can we point to a famous mathematician that came up with the theory of "constructions" by limits? Or is this just something that kind of crept into mathematics on the sly?

I mean, I'd really like to know just who it was that justified the idea that Weierstrass's limit definition can be used to claim that a mathematical object has been constructed in its entirety.

This isn't merely a philosophical question. This is a problem with logic. As I mentioned in an earlier post Weierstrass included in his limit definition the condition that delta cannot equal zero,… i.e.:

\exists\delta\ni\delta&gt;0

This little piece of information is not trivial. In fact, it's there because without it the rest of the definition would fall apart. It's crucial to the definition that delta not be allowed to become zero. This a very important part of the logic that insures that conclusions of the definition are meaningful. Take that little piece of the definition out and the resulting conclusions have no foundation in logic.

Well,… ignoring that little piece of the definition is precisely what we must do if we want to claim that we have completed the limit process and constructed a complete mathematical object in its entirety. Yet this is precisely what mathematicians are doing when they claim to have constructed a mathematical objecting using the definition of the limit.

Therefore I hold that those constructions are logically flawed and therefore they cannot be depended upon to be logically meaningful in any way.

To say that 0.9999…. = 1 in an absolute way is to say that we have taken delta to equal zero in Weierstrass's limit definition and that's a violation of logic. We are simply logically incorrect to make such a claim. We have no logical justification for doing so.

This is much more than merely a philosophical opinion. This is a serious logical issue.

 

[matt grime]

If the reals are not constructed as either dedekind cuts or cauchy sequences, what are they? as a model of a complete totally ordered field I mean.

Things are constructed in mathematics all the time. Arguably everything in mathematics is a construction in some sense. Are you positioning yourself as a platonist and claiming that there is a physical object that is the real numbers? If so what is it? Maths generally isn't done like that. It is merely a formal thing we play around with. If it can be usefully used to model the real world so much the better, I suppose, but no one should actually think that the things we use in maths have any existential form. It is not necessary, and frequently not useful.

 

[StatusX]

Constructed it as a limit?

This is the kind of talk that just blows me away.

Where's the justification for claiming that anything has been "constructed"?

...

This isn't merely a philosophical question. This is a problem with logic. As I mentioned in an earlier post Weierstrass included in his limit definition the condition that delta cannot equal zero,… i.e.:

\exists\delta\ni\delta&gt;0

This little piece of information is not trivial. In fact, it's there because without it the rest of the definition would fall apart. It's crucial to the definition that delta not be allowed to become zero. This a very important part of the logic that insures that conclusions of the definition are meaningful. Take that little piece of the definition out and the resulting conclusions have no foundation in logic.

Well,… ignoring that little piece of the definition is precisely what we must do if we want to claim that we have completed the limit process and constructed a complete mathematical object in its entirety. Yet this is precisely what mathematicians are doing when they claim to have constructed a mathematical objecting using the definition of the limit.

Therefore I hold that those constructions are logically flawed and therefore they cannot be depended upon to be logically meaningful in any way.

To say that 0.9999…. = 1 in an absolute way is to say that we have taken delta to equal zero in Weierstrass's limit definition and that's a violation of logic. We are simply logically incorrect to make such a claim. We have no logical justification for doing so.

This is much more than merely a philosophical opinion. This is a serious logical issue.

I understand what your saying, but youre missing something important. All I want you to do is tell me which step has the mistake:

1.
0.999... \equiv \lim_{n \rightarrow \infty} \sum_{k=1}^n 9 \cdot 10^{-k}

That triple equal sign means "is defined as."

2.
\lim_{n \rightarrow \infty} \sum_{k=1}^n 9 \cdot 10^{-k} = L

If there is a number L such that for any e>0, you can find an n such that |L - S(n)| < e, where S(n) is the partial sum of the first n terms.

3.
In this case, it is easy to prove that n = floor(2 - log(e)) will satisfy the condition |1 - S(n)| < e for any e>0, and so L=1.

4.
So, by the transitivity of equality:

0.999... = 1

I know this 0.999... thing is getting tiring, but this will help you understand why your wrong. And please get back to reality, you are not discovering some hidden flaw no one has seen before. You are misunderstanding basic concepts about real numbers, the way they were defined.

 

[NeutronStar]

I understand what your saying, but youre missing something important. All I want you to do is tell me which step has the mistake:

Well obviously your very first step is logically incorrect.

rewrite it as follows and you'll see why,…

f(c) \equiv \lim_{x \rightarrow c} f(x)

You simply have no logical basis for defining something as its limit. f(x) may not have a value at f(c). You can't use the definition of the limit to define the value f(c). There's nothing in the definition of the limit that supports this.

Just because 0.999… has a limit defined at infinity doesn’t' mean that it is equal to that limit.

There is nothing in the definition of a limit that allows you to claim that any process actually equals its limit.

On the contrary, the part of the limit definition that says,…

\exists\delta\ni\delta&gt;0

actually forbids you from claiming that equality using the defintion of the limit alone.

So not only are you incorrect in doing this, but you are actually forbidden by definition to do it.

If mathematicians are doing this on a regular basis then all they are really doing is ignoring the details of Weierstrass's definition.

 

[CrankFan]

Constructed it as a limit?

This is the kind of talk that just blows me away.

Where's the justification for claiming that anything has been
"constructed"?

There are many equivalent ways to define real numbers, one method that's
used a lot is the Cauchy construction. A real number can be thought of
as an equivalence class of Cauchy sequences of rational numbers.

What's a Cauchy sequence, you say? I'm glad you asked!

A Cauchy sequence of rational numbers is a sequence x of
rationals such that for every positive rational number \epsilon <br />
there exists a positive integer N such that for
every m, n &gt; N we have:

\vert x_n - x_m \vert &lt; \epsilon

In English, for any epsilon (no matter how small!) there is some point
in the sequence, after which the difference of any two terms is less
than that epsilon.

In this construction reals aren't actually Cauchy sequences of
rationals, but equivalence classes of Cauchy sequences of
rationals, and this is how we get to .999... = 1

1, 1, 1, 1, 1, 1, ... is a Cauchy sequence of rationals. You can
confirm this with the definition provided above.

9/10, 99/100, 999/1000, ... is also a Cauchy sequence of rationals,
again you can confirm it based on the definition above.

So the question you might be thinking about at this point is how is the
equivalence relation on Cauchy sequences of rationals defined? I'm glad
you asked!

We say that two Cauchy sequences x and y are equivalent iff
for every positive rational number \epsilon there is an
integer N such that for all n &gt; N we have:

|x_n - y_n | &lt; \epsilon

In other words, for every epsilon greater than zero (no matter how small!)
there is a point in both sequences after which the difference between any
two terms is less than that epsilon.

Recall the two Cauchy sequences in question:

x = 1, 1, 1, 1, 1, 1, ...
y = 9/10, 99/100, 999/1000, ...

To prove they are equivalent we must show that for every \epsilon
there is an N such that n &gt; N implies |x_n - y_n | &lt; \epsilon

|x_n - y_n | = 1^n - { (10^n - 1) \over 10^n } = { 1 \over 10^n }

If we choose an N such that 10^N &gt; { 1 \over \epsilon } then we have

|x_n - y_n | &lt;= { 1 \over 10^n }

|x_n - y_n | &lt; { 1 \over 10^N }

|x_n - y_n | &lt; \epsilon

And we're done. (Uh, I think... pending any corrections from mentors )

 

[Hurkyl]

I'm sure that Euclid would roll over in his grave if he knew that his dimensionless points had gone off their diets.

I'm not sure whose comment you were addressing, but it wasn't mine. I was addressing your comment that "It's simply a logical contradiction in terms to claim that an infinite process can be completed."

How then can such a point be divisible?

Nobody says a point is divisible. Divisibility refers to space.

I was suggesting that in reality .999... does not equal 1, it equals .999..

Just like 1/2 doesn't equal 2/4, I suppose.

Well obviously your very first step is logically incorrect.

rewrite it as follows and you'll see why,?

f(c) \equiv \lim_{x \rightarrow c} f(x)

This is exactly your misconception, because this is exactly what StatusX was not saying.

 

[StatusX]

You simply have no logical basis for defining something as its limit.
...
Just because 0.999… has a limit defined at infinity doesn’t' mean that it is equal to that limit.
...
There is nothing in the definition of a limit that allows you to claim that any process actually equals its limit.

I'll try to be extremely careful and thorough here so if there are any errors they can be easily spotted.

Decimal notation is defined in terms of limits. A decimal expansion consists of an infinite series of integers between 0 and 9:

{d_N, d_{N-1}, ..., d_1, d_0, d_-1,...}

In general, this starts at some integer N and goes to -\infty. The real number r represented by this series is defined as:

r \equiv \sum_{n=-\infty}^{N} d_n \cdot 10^n

If you want to get really technical, you can rewrite this (yea, this is just a rewrite) as:

r \equiv \lim_{m \rightarrow \infty} \sum_{n=-m}^{N} d_n \cdot 10^n

This is a definition. ok? If you dispute the truth of this statement, then you arent talking about the same decimal notation as the rest of us. If you don't think this definition is logically sound, I'll address that below.

Now, you might argue that irrational numbers are poorly defined in this system, but I'm ignoring them for now. Any repeating decimal can be rewritten as an infinite sum, or again, if you are fussy, as the limit of a sequence of partial sums. For example, for 0.333...(where the dots just imply that d_m=3 for any arbitrarily large negative integer m):

\lim_{m \rightarrow \infty} \sum_{n=-m}^{-1} 3 \cdot 10^n

= \lim_{m \rightarrow \infty} \sum_{n=1}^{m} 3 \cdot 10^{-n}

= 1/3

I could show this using the epsilon delta defintion, but I hope you don't need me to. So, just to reiterate: 0.333... is a mathematical symbol, just like an integral sign or a radical. It is defined as the value of a limit, which is in turn defined by the epsilon delta method.


Now your problem, which is demonstrated nicely in the quotes above, is that you are confusing the value of a limit with the process of taking partial sums. These are not the same. There are no variables in 0.999...: it is a constant, and it is meaningless to take a limit of it.

What you are implicitly assuming is that the number has to be written out completely to have a value. If you sat down with a pen and paper and started writing "0." followed by as many nines as you could, you would be performing a process. The number you write down would never equal 1, no matter how many nines you write (note that infinity isn't a number, not to mention the universe is finite). However, this is NOT what 0.999... means in ANY sense. The abstract mathematical symbol 0.999... is defined as above, and is equal to 1.

to be clear:
A zero, followed by a decimal point, followed by three nines, followed by three dots is an abstract symbol which is meant to reperesent the value of an infinite sum, or more precisely, the limit of a sequence of partial sums, which in this case turns out to be equal to 1.

It is very important you understand the difference between the process of listing numbers and taking a limit. When you say something that I can only interpret as "limits can never exactly equal their limit, they just approoach it," you are talking nonsense. Specifically, by the first "limits," you mean the partial sums, or the values of a function as x gets closer and closer to c. It is true, these never equal the limit value, but they are completely separate entities from this value. The limit is the number L as defined in the epsilon delta method. By this definition, none of the partial sums or close values are equal to it. But these are only used in calculating the limit. L is a real number, and it is not changing in any sense.


One final point. You mentioned that I can't say that:

\lim_{x \rightarrow c} f(x) = f(c)

This is true in general. However, in this case, c is infinity, and in this case, that statement is true. In fact, it's how infinity is defined!

f(\infty) = \lim_{x \rightarrow \infty} f(x)

There is no number infinity, so it must be treated as special, as in this example. Also, infinite limits are different in ordinary limits because instead of getting closer and closer to some value, x is alowed to get bigger and bigger without bound. Again, infinity is only definined in the context of limits.